Given a real example for each of dirac's 4 postitulates.
1. the state of a physical system is given by the data of a ket (or vector) latex: \left|\psi\right\rangle. latex: \left|\psi\right\rangleis called the quantum state and can be decomposed linearly on a base latex: \lbrace \left|\phi_n\right\rangle \rbrace.
2. any observable x is described by an operator latex: a (with eigenvalues latex: a_n and eigenvectors latex: \left| n\right\rangle
3. the measurement of the observable x can only yield one of the eigenvalues latex: a_n of the operator latex: a.
4. if we measure the observable x of a system in a normalized quantum state latex: \left|\psi\right\rangle, the probability latex: \mathcal{p}(a_n) of obtaining the eigenvalue latex: a_n is latex: \mathcal{p}(a_n)=\left| \langle n|\psi \rangle \right|^2